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This case includes all defect
situations that one could find in a perfect ionic crystal. Besides Schottky and
Frenkel disorder or any mixture of the
two, we could have many more defects - any combination of interstitials and
vacancies for any kind of atom in the crystal is admissible. |
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One example: In an AB crystal, instead if two
oppositely charged vacancies (the Schottky defects), we could also have two
interstitials of the two kinds of atoms carrying different charge. |
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This kind of defect is called an "Anti Schottky defect", it would be
formed according to the (Kröger-Vink) reaction |
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Following our
recipe, we obtain after rearranging and "translating" to Schottky
notation: |
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(Ai·
Vi) + (Bi/
Vi) |
= |
AB |
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This is essentially the same result
as for the regular Schottky defects. However, in using the mass action law, we
would have to use the formation energies for interstitials (and take care of
the additional degrees of freedom for arranging interstitials
as in the case of Frenkel
defects). |
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In reality, the formation energies of interstitials are mostly
larger than those for vacancies; this is certainly true for the "big"
negatively charged anion interstitial. |
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Anti Schottky defects therefore have not been observed as the
dominating defect type so far. But they are not only perfectly feasible, but
also always present - only their concentration is so small that it can not be
measured; consequently they do not play a role in anything interesting
connected with defects. |
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We could also
conceive of "anti site
defects", i.e. A-atoms on B-places and vice versa;
AB and BA, of combinations like a
VA and AB, and of plenty more intrinsic
defects for more complicated crystals, e.g. for
YBa2Cu3O7 (the famous first high
temperature superconductor with a critical temperature larger than the boiling
point of liquid N2). |
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We could, moreover, include isoelectronic impurity atoms into
the list; e.g. K instead of Na; Ba instead of Ca,
or F instead of Cl, which could be incorporated into a crystal
without the need to change anything else. A little dirt, after all, is always
"intrinsic", too. |
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And all
the reactions that are conceivable will occur. The only difference is that some might be
frequent and some might be rare - and it is often sufficient to only consider
the dominating reaction. |
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This teaches us a
major lesson, especially with respect to
the upcoming paragraphs |
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There are far too many defect types and
reactions principally possible in simple (ionic) crystals (not to mention
complicated ones) for a priori treatments of all possible effects. We must
invoke some additional information (such as
anion interstitials being unlikely) to simplify the situation to a level where
it can be handled. |
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For the intrinsic defects mentioned so far, this was easy and
has been done all along. It will become an important guiding principle for the
other two cases, however. |
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But we are not yet
done with intrinsic defects: If we look at semiconducting ionic or compound crystals, we may
have to include electrons and holes in our
defect systematization. Let's look at this, first assuming that electrons and
holes are the only defects. |
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Implying a
band structure and
always using the Boltzmann approximation for the tail of the proper Fermi
distribution, we might denote the generation of an electron in the conduction
band in complete analogy to the Kröger-Vink system.by |
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Rearranging gives |
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(eC/
h·C) +
(h·V e/V)
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= |
0 |
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The obvious translation to Schottky notation yields |
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Adding electroneutrality, i.e. e/ = h·, makes the
analogy to Frenkel defects complete,
and we obtain |
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There is a big difference, however.
We know
that the constant c in front of the exponential is the effective
density of states Neff (or more precisely
[NCeff ·
NVeff]½) and the formation
energy is given by E = Eg/2, with
Eg = band gap. |
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This is knowledge that comes from quantum
theory and there is simply no way to deduce
this from classical thermodynamics. |
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However, the mass action law is based
on considering the minimum of the free enthalpy - a principle that is always
valid. It is only the chemical potential of electrons and holes that cannot be
directly expressed in the standard form. Mass action, however, remains
valid. |
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Lets now consider some typical doping
reactions. Most common and important is the doping of semiconductors with
substitutional impurities, i.e. P, As, or B in
Si. |
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If we call the substitutional dopant atoms D (for
Donor) or A (for Acceptor), we may express the doping
reaction, i.e. the exchange of electrons or holes with the bands, as
follows |
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Kröger-Vink and Schottky notation are identical in this
case (figure it out!), and we have the mass action law |
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Lets consider a simple situation with just one type of doping,
say donors, with a concentration [D], giving us [e/] =
kD1[D]/[D·] as stated above. But we have no holes so
far. We need some other reaction to produce holes, what comes to mind is |
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Can we get the "universal"
mass action law for semiconductors, as we know it from semiconductor physics
from this, i.e. |
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If we form the product with the equation from above we obtain
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We have the mass action law as we know it with, however,
unspecified constant. In order to obtain the absolute concentrations, we need
one more equation, which, in the absence of other charged defects, is supplied
by the electroneutrality condition |
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OK, we are on safe grounds again. But
there seems to be a certain ambiguity. Instead of the reaction
h· + D = D·, we
also could have chosen the "normal" intrinsic reaction
e/ + h· = 0 from above. |
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So what is it? Both, of
course. The equations above are dominating at low temperatures where thermal
carrier generation can be neglected, i.e. not too high temperatures, the other
one dominates at high temperatures. |
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But both occur independent
of each other and, since there is only one
equilibrium value for the respective concentrations, both must give the same numerical values for the
same quantity if evaluated. |
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This teaches us an important lesson
for the treatment of defect equilibria: |
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Since the mass action law
and the electroneutrality condition supply only two equations for possibly more than two unknown
defect types, any sensible reaction
equation that comes to mind and contains the unknown quantities can be used to
supply the required additional information! The equilibrium concentration of
defect type i is always the same - no matter in which equation it comes
up! |
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As we see, even for pure
semiconductors it is possible to describe the electron-hole equilibrium in
terms of reaction equations. |
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But the notion of chemical potentials becomes
somewhat strained for calculating concentrations or "activities".
Considering densities of states and distribution functions (Fermi distribution
in full generality or Boltzmann distribution in the proper approximation) may
be more advantageous as long as only electrons and holes are considered. |
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Now lets look at a different kind of
doping: We intentionally change the vacancy concentration, i.e. we
dope a crystal with vacancies. |
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In a simple example we may look back at the introductory paragraph of this subchapter, and
consider a reaction equation for the incorporation of Ca into a
NaCl crystal (in Schottky notation right away - can you figure out the
Kroeger-Vink notation?) |
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In words: A Ca-VNa pair is introduced (or
taken out ) of the crystal. |
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The mass action law demands that
[Ca·Na] · [V/Na] =
constant; and charge neutrality is conserved if [V/Na]
= [Ca·Na]. |
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If, and only if there is
no other way to achieve charge neutrality, e.g. by generating electrons or
holes, we will now produce vacancies by
incorporating a doping element in perfect analogy to producing electrons or
holes in semiconductors. |
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Of course, we could also have
incorporated Ca by generating Cl/ interstitials, a mix
of vacancies and interstitials, or even worse, a mix of vacancies,
interstitials, holes and electrons. All those possible reactions will
occur and we cannot know a priori what will dominate. In a real case we must
use some additional knowledge as pointed out
above if we want to get results. |
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Well, "we" do know that vacancy doping can indeed be
achieved in this way in ZrO2 doped with e.g.
Y2O3 or CaO, generating
VO··, i.e. doubly positively charged
oxygen vacancies. This is a particularly relevant example, because it is part
of the working principle of the oxygen sensor in your car exhaust system that
feeds the controller of the car engine in order to keep emissions at the lowest
possible level. |
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The technical importance is the same as in semiconductors:
Whatever the intrinsic defect concentrations might be in the perfect intrinsic
material, with doping you have a more or less fixed concentration of vacancies
that can be far larger than the intrinsic concentration and may not depend
sensitively on temperature. |
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Great if you need lots of vacancies
because you want to make an ionic conductor where the conductivity depends on
the diffusion of oxygen via a vacancy mechanism. However, vacancy doping is not
such a hot issue at low temperatures (like room temperature) if the vacancies -
and therefore the oxygen ions, too - are not mobile at reasonable temperatures
- in contrast to electrons and holes which get rather more mobile with
decreasing temperature. |
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But if there is some
mobility, you have now increased the ionic
conductivity by orders of magnitude - exactly as you increase the electronic conductivity in semiconductors by
doping. |
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Moreover, you know the
concentration of the vacancies, and within some parameter range, you can treat
it as constant which means you can remove
[V] from the business end of the mass action law and multiply it into
the general reaction constant. Life is easier. |
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There are more examples for technical
uses of doping, but we will now consider the third basic reactions, the defect reactions at
interfaces. |
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Considering that all chemical reactions between a solid and anything
else occur at the surface of the solid, i.e. at the solid-gas, solid-liquid, or
solid- solid interface, this headline covers a good part of general
chemistry. |
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Indeed, it has become clear in recent years that reactive
solid-solid interfaces generate or consume point defects. However, here we will
only look at reactions between a gas and a (simple) ionic crystal as they are
used in sensor technology. In other words, we consider the possible reactions
between a MeX crystal and a X2(g) gas in a first
simplified treatment. We do not consider charges for the time being, to keep
things simple. |
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The crystal may then incorporate X2(g) (or
emit it) via several reactions which we can easily formulate with the the
Kröger-Vink notation: |
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i.e. an atom of the gas occupies a fitting empty place (=
vacancy) of the crystal. |
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i.e. an atom of the gas occupies a fitting empty place (=
vacancy) in the interstitial lattice and is now an interstitial. |
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i.e. an atom of the gas occupies a (probably not well
fitting) empty place (= vacancy) in the metal ion lattice and is now an
anti-site defect. |
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And there will be even more possibilities as is shown
below. |
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The thing to note
once more is: If these reactions can occur, they will occur - independently of each other. Only their
probabilities (or reaction rates) are (wildly) different, and we are well off
if we know (or can make an educated guess) at the dominating reaction. And the
equilibrium concentrations [Di] of some defect type
Di, no matter in which equation it appears, are
always identical in equilibrium. |
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In other words, we should know
- What kind of defect situation dominates in our MeX crystal (Frenkel-
or Schottky defects etc.); i.e. which reaction constant is smallest?
- How can charge neutrality be achieved (only with ions and defects; only
with electrons and holes, or in a mixture)? In other words are we dealing with
an ionic conductor, a semiconductor, or a mixed case? Of course, the answer to
this question may well depend on the temperature.
- Is there intentional (or unintentional) ionic or electronic doping that
imposes specific conditions on the defect situation?
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This only looks hopeless, but
rejoice, it is not - albeit for a sad reason: After all, we are not so much
interested in defects per se, but in their uses. Typically, we want to do
something "electrical" - make a better battery, a fuel cell, a sensor
giving of a voltage or a current in response to the stuff to be sensed - and
this demands that we use only ionic crystals that are
ionic conductors of some sort. |
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Unfortunately, not too many ionic crystal are useful ionic
conductors; it's just a handful of crystal families. And only those families we
have to know. The search for a real good (and affordable) ionic conductor at
room temperature is still on - if you find it, you may not make the Nobel
prize, but certainly a great deal of money. |
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So all it takes is to study some 4 or 5 typical
cases which contain practically everything encountered in ionic defect
engineering. |
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The paradigmatic case is an undoped
crystal with Frenkel defects and some semiconducting properties. We start
assuming that the electron/hole concentration is far smaller than the Frenkel
defect concentration. |
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Thermal equilibrium of the MeX crystal
by itself thus means that we have |
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Now we establish equilibrium with the
gas X2(g). It could be H2,
O2, Cl2, F2,
whatever. |
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How are X atoms to be incorporated? Surprisingly,
perhaps, none of the three possibilities given above is the preferred reaction.
We do not have X- vacancies (Vx) available in our
case, and we are not going to generate very unlikely X interstitials
(Xi') or anti site defects. We want to incorporate X
on a regular lattice site. |
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Now you see why the Kröger-Vink notation
is useful. Playing around a bit with what you have (and putting in charge
neutrality right away), gives |
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½ X2(g) + MM |
= |
MX + V/M +
h· |
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In words: An X-atom takes out a M atom from its
position on the crystal surface, forming a MX molecule that is added to
the crystal somewhere, leaving back a vacancy on a M-site and a
hole. |
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This is not so easily expressible in Schottky notation (try it), but leads easily to the mass
action law noticing that [MM] = [MX] = const =
1 and thus not needed in the "business end" of the mass action
law. |
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We could come up with other reaction equations achieving the
same result; but the one we have is good enough for the time being. |
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We thus obtain two "master"
equations for this case, one from mass action and one from charge neutrality
exactly along the lines discussed
before. |
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Mass action law |
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[X2(g)]1/2
[V/M] · [h·] |
= |
const. |
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Charge neutrality |
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[V/M] + [e/] |
= |
[Mi·] + [h·] |
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We have two equations for
the four unknowns V/,
e/, Mi·, and
h· which determine the electrical conductivity σ as a function of the concentration (or partial pressure) of
the gas [X2(g)] via |
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σ([X2(g)]) |
= |
Σi (qi ·
ci · μi) |
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With q = charge carried by the defect i,
ci = concentration and μi = mobility of defect i,
resp. |
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Again, we need to have additional information about our system
if we want quantitative relations between a measurable parameter like the
defect-dependent conductivity. |
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Doping, as described above, could be helpful. It would provide
a more or less constant value for e.g. the vacancy or electron concentration
and thus remove one (or two) unknowns. |
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Without that, however, we have to resort to case studies
making reasonable assumptions and considering the important quantities for the
task at hand. As an example, if we want to measure the
[X2(g)] concentration, we are not so much interested in the
absolute value of σ, but in its change with
the gas concentration, dσ/d[X2(g)]. |
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That implies that we are mainly interested in those defects
which react sensitively to concentration changes of
[X2(g)]. |
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For our example
we postulated the two conditions |
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This is valid as long as we are considering
stoichiometric MX which is neither losing nor adding X. In other
words, the crystal is kept at the stoichiometric point - at a certain
partial pressure of X2. |
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It is now important to notice that the reaction
with the outside gas at partial pressures different from that belonging to the
stoichiometric point changes the stoichiometry - no matter how you look at it.
For ambient (or standard) pressure, there is no reaction and the stoichiometry
is perfect - we are at the stoichiometric point. For large partial pressures of
X2, we will produce MX1 +d, , for low pressures MX1
d, . |
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Now lets see what happens if we work
around the stoichiometric point. The absolute concentration of the vacancies
and interstitials may change a little, and this means that the electron and
hole concentration has to change exactly the same amount in order to maintain
charge neutrality. |
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However, since we assumed that the absolute
concentration of the electrons and holes at the stoichiometric point is much
smaller than that of the vacancies and interstitials, the relative change of [e/] and
[h·] is much larger than that of
[Mi·] and
[V/M]. |
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Accordingly, we may assume that
[Mi·] and [V/M]
» constant around the
stoichiometric point, i.e. within a certain range of partial pressures of
X2 below and above the standards pressures which simplifies
the relevant equations to |
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With a different value of the constant,
however. |
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The concentration of holes and electrons, on the
other hand, changes markedly, but their absolute concentrations are still much smaller than
that of vacancies and interstitials. |
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This leads us to an
extremely important generalization: |
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As far as the mass action law is concerned, only
variable concentrations, i.e.
concentrations that are not (approximately) constant, count. The absolute
concentration is of no special importance - it becomes part of the reaction
constant. |
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As far as charge neutrality is concerned, only absolute concentrations count. Minority carriers can
simply be neglected in a first approximation. |
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This give a first direct result: We
can write dow the following simplified mass action law and electroneutrality
condition: |
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Mass
action |
[h·] |
= |
const · [X2(g)]1/2 |
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Electro-
neutrality |
[V/M] |
= |
[Mi·] |
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How about the electron concentration?
Since our approximations imply that there is no interaction between the point
defects and the electrons and holes, we must have [h·] · [e'] = const. and thus |
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[e/] |
= |
const · [X2(g)]1/2 |
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But we will derive that result now by using
different reaction equation just to show that in equilibrium you always must
obtain the same results. |
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Lets consider the reaction |
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½ X2(g) + M·i
+ e/ |
= |
MX + Vi |
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In words: A positively charged metal interstitial
plus an electron and a gas atom form a crystal molecule and a vacancy on the
interstitial lattice. |
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Neglecting M·i,
MX, and Vi with the same arguments as before, we have,
as we know that it must be |
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[e/] |
= |
const · [X2(g)]1/2 |
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In a log [i] - log
[X2(g)] plot we have straight lines with a slope of 1/2
for holes, 1/2 for electrons and 0 for the interstitials
and vacancies, respectively. This looks like this: |
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Without looking at the reaction constants, we
know that the cross over of the e/ and
h· concentration lines must be at the stoichiometric
point. |
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It is clear that for large deviations from the
stoichiometric point the approximations used are no longer valid. For very
small or very large partial pressures of X2, we now may
consider the other two possible extremes by simply extrapolating the lines in
the illustration: |
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1. For very large partial pressures of
X2 the e/ concentration becomes negligible
while the hole concentration becomes comparable to the point defect
concentration. Charge neutrality can only be maintained by decreasing the
positively charged metal interstitials and increasing the negatively charged
vacancies. In the extreme, we may only consider [V/M]
= [h·] for charge neutrality. Inserting that in the
reaction equation from above, we have . |
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which gives the mass action equation |
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[X]1/2
[h·]2 |
= |
const |
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[h·] |
= |
const. · [X]1/4 |
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For very low partial pressures we obtain exactly
along the same line of arguments [e/] = const. ·
[X]1/4. |
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With these relations, we may also
calculate the concentrations of the minority point defects by simply inserting
the above equations in the appropriate reaction equation and applying mass
action which yields |
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High pressure
side |
[Mi·] |
= |
const · [X]1/4 |
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Low pressure
side |
[V/M] |
= |
const · [X]1/4 |
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Putting everything together in on single graph,
we obtain a schematic Kröger-Vink
or Brouwer
diagram: |
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Of course, the change-over would be smooth in
reality; and we cannot tell easily where it will occur. It is also obvious that
there are no discontinuities of the concentration curves, which tells us
something about the "const." in the mass action equation. |
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In the consideration above we did not assign
values to the "const." and carry it through. That might be a
interesting exercise one of those days. |
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In any case, we now have seen how
Kröger-Vink reaction equations, mass action, charge neutrality and some
additional knowledge or educated guesses allow to come up with a pretty good
idea of what will happen in a reaction involving point defects. |
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The possibilities of electronic and ionic doping
together with the temperature dependence of point defect equilibria now give us
a powerful tool for designing materials for specific applications. |
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"Ionics",
in research and application, is slowly coming into its own. Together with the
good old "electronics" it may well open up new fields for materials
scientists. |
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© H. Föll